Approximation properties relative to continuous scale space for hybrid discretizations of Gaussian derivative operators
Tony Lindeberg
TL;DR
This work analyzes two hybrid discretisations for Gaussian derivative operators—one using a normalised sampled Gaussian kernel and the other using an integrated Gaussian kernel—paired with central differences, and compares them against fully continuous and genuinely discrete discretisations. It introduces quantitative metrics for spatial smoothing (via $\sqrt{V(|T_{x^{\alpha}}(\cdot;s)|)}$) and scale-estimation accuracy (via $O_{\alpha}(s)$ and $E_{\text{scaleest,rel}}(\sigma)$) to assess how well discretised operators preserve scale-space properties, especially at very small scales where discrepancies arise. The methodology extends scale-space analysis to 2-D with separable implementations and evaluates scale-selection detectors (Laplacian, Hessian, gradient magnitude, and ridge) under automatic scale selection, revealing trade-offs: hybrids offer computational efficiency but can diverge from continuous theory at fine scales, while the discrete analogue kernel generally aligns best with continuous expectations. The findings guide discretisation choices for multi-scale Gaussian-derivative tasks, including deep-learning contexts where primitive kernels like modified Bessel functions may be unavailable, and provide practical code in the pyscsp package for reproducibility and integration into scale-space or deep learning pipelines.
Abstract
This paper presents an analysis of properties of two hybrid discretization methods for Gaussian derivatives, based on convolutions with either the normalized sampled Gaussian kernel or the integrated Gaussian kernel followed by central differences. The motivation for studying these discretization methods is that in situations when multiple spatial derivatives of different order are needed at the same scale level, they can be computed significantly more efficiently compared to more direct derivative approximations based on explicit convolutions with either sampled Gaussian kernels or integrated Gaussian kernels. While these computational benefits do also hold for the genuinely discrete approach for computing discrete analogues of Gaussian derivatives, based on convolution with the discrete analogue of the Gaussian kernel followed by central differences, the underlying mathematical primitives for the discrete analogue of the Gaussian kernel, in terms of modified Bessel functions of integer order, may not be available in certain frameworks for image processing, such as when performing deep learning based on scale-parameterized filters in terms of Gaussian derivatives, with learning of the scale levels. In this paper, we present a characterization of the properties of these hybrid discretization methods, in terms of quantitative performance measures concerning the amount of spatial smoothing that they imply, as well as the relative consistency of scale estimates obtained from scale-invariant feature detectors with automatic scale selection, with an emphasis on the behaviour for very small values of the scale parameter, which may differ significantly from corresponding results obtained from the fully continuous scale-space theory, as well as between different types of discretization methods.